The Biot-Savart law is a key principle in electromagnetism, describing how electric currents generate magnetic fields. It provides a mathematical framework for calculating the strength and direction of magnetic fields produced by current-carrying wires of any shape.

This law is crucial for understanding the relationship between electricity and magnetism. It forms the foundation for designing electromagnetic devices and helps explain various magnetic phenomena observed in nature and technology.

Magnetic field of current-carrying wire

• Current-carrying wires generate magnetic fields in the space surrounding them
• The strength and direction of the magnetic field depend on the magnitude and direction of the current
• Understanding the magnetic fields produced by current-carrying wires is crucial for designing electromagnetic devices (transformers, motors, generators)

Biot-Savart law

Mathematical formulation

• Biot-Savart law is a mathematical expression that relates the magnetic field generated by a current-carrying wire to the current and the geometry of the wire
• The law is given by the equation: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$
  • $d\vec{B}$ is the infinitesimal magnetic field
  • $\mu_0$ is the permeability of free space
  • $I$ is the current
  • $d\vec{l}$ is an infinitesimal length element of the wire
  • $\hat{r}$ is the unit vector pointing from the current element to the point where the magnetic field is being calculated
  • $r$ is the distance between the current element and the point where the magnetic field is being calculated

Infinitesimal current element

• The Biot-Savart law considers the magnetic field contribution from an infinitesimal current element $d\vec{l}$
• The infinitesimal current element is a small segment of the current-carrying wire
• By integrating the contributions from all the infinitesimal current elements along the wire, the total magnetic field can be determined

Cross product in equation

• The Biot-Savart law involves a cross product between the infinitesimal current element $d\vec{l}$ and the unit vector $\hat{r}$
• The cross product determines the direction of the magnetic field according to the right-hand rule
  • Point the thumb of your right hand in the direction of the current
  • Curl your fingers around the wire
  • The direction your fingers point is the direction of the magnetic field
• The magnitude of the cross product is proportional to the sine of the angle between $d\vec{l}$ and $\hat{r}$

Applications of Biot-Savart law

Magnetic field of straight wire

• The Biot-Savart law can be used to calculate the magnetic field around a straight current-carrying wire
• The magnetic field lines form concentric circles around the wire
• The magnitude of the magnetic field at a distance $r$ from a long straight wire is given by: $B = \frac{\mu_0 I}{2\pi r}$
  • $\mu_0$ is the permeability of free space
  • $I$ is the current in the wire
  • $r$ is the distance from the wire

Magnetic field of circular loop

• The Biot-Savart law can be applied to calculate the magnetic field at the center of a circular current loop
• The magnetic field at the center of a circular loop is given by: $B = \frac{\mu_0 I}{2R}$
  • $\mu_0$ is the permeability of free space
  • $I$ is the current in the loop
  • $R$ is the radius of the loop
• The magnetic field lines inside the loop are uniform and perpendicular to the plane of the loop

Magnetic field of solenoid

• A solenoid is a long, tightly wound coil of wire
• The Biot-Savart law can be used to calculate the magnetic field inside a solenoid
• The magnetic field inside an ideal solenoid is uniform and parallel to the axis of the solenoid
• The magnitude of the magnetic field inside a solenoid is given by: $B = \mu_0 n I$
  • $\mu_0$ is the permeability of free space
  • $n$ is the number of turns per unit length of the solenoid
  • $I$ is the current in the solenoid

Derivation of Biot-Savart law

Lorentz force law

• The Biot-Savart law can be derived from the Lorentz force law
• The Lorentz force law describes the force experienced by a moving charge in the presence of electric and magnetic fields
• The magnetic force on a current-carrying wire can be obtained by integrating the Lorentz force over the length of the wire

Magnetic field vs electric field

• The Biot-Savart law specifically deals with the magnetic field generated by current-carrying wires
• The electric field, on the other hand, is related to the presence of electric charges
• The Biot-Savart law and Coulomb's law (for electric fields) have similar mathematical forms, but they describe different phenomena

Superposition principle

• The Biot-Savart law relies on the superposition principle
• The superposition principle states that the total magnetic field at a point is the vector sum of the magnetic fields contributed by each current element
• This allows the magnetic field contributions from different parts of a current-carrying wire to be added together to determine the net magnetic field

Limitations and assumptions

Steady current approximation

• The Biot-Savart law assumes that the current in the wire is steady (constant in time)
• If the current varies with time, the magnetic field will also vary, and the Biot-Savart law may not accurately describe the field

Magnetostatic fields

• The Biot-Savart law is applicable to magnetostatic fields, which are magnetic fields that do not change with time
• In situations where the magnetic field is time-varying (such as in electromagnetic waves), the Biot-Savart law needs to be modified or replaced by more advanced formulations (Maxwell's equations)

Validity at atomic scales

• The Biot-Savart law is a classical description of magnetic fields and assumes that the current is continuous
• At atomic scales, where quantum effects become significant, the Biot-Savart law may not provide an accurate description of the magnetic field
• In such cases, quantum mechanical treatments (quantum electrodynamics) are necessary to describe the magnetic interactions

Ampère's circuital law vs Biot-Savart law

Calculating magnetic fields

• Both Ampère's circuital law and the Biot-Savart law can be used to calculate magnetic fields generated by current-carrying wires
• Ampère's circuital law relates the magnetic field around a closed loop to the current enclosed by the loop
• The Biot-Savart law, on the other hand, calculates the magnetic field by considering the contributions from infinitesimal current elements

Simplicity of Ampère's law

• Ampère's circuital law is often easier to apply when the system has a high degree of symmetry (infinite wires, solenoids, toroidal coils)
• In such cases, Ampère's law can provide a more straightforward way to calculate the magnetic field compared to the Biot-Savart law

Generality of Biot-Savart law

• The Biot-Savart law is more general and can be applied to any current distribution, regardless of the geometry
• It is particularly useful when the current distribution is complex or lacks symmetry
• The Biot-Savart law can be used to derive Ampère's circuital law, demonstrating its fundamental nature in the description of magnetic fields